Quasi-geostrophic equations with initial data in Banach spaces of local measures

This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations $$displaylines{ partial_{t}heta+u ablaheta+( -Delta) ^{gamma}heta =0 quad hbox{on }mathbb{R}^{d}imes] 0,+infty[cr heta( x,0) =heta_{0}(x), quad xinmathbb{R}^{d} }$$ where 0 less than $gammaleq 1$ is a fixed parameter and the velocity field $u=(u_{1},u_{2},dots,u_{d}) $ is divergence free; i.e., $ abla u=0)$. The initial data $heta_{0}$ is taken in Banach spaces of local measures (see text for the definition), such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case 1/2 less than $gammaleq1$ and we analyse the well-posedness of the system in three basic spaces: $L^{d/r,infty}$, $dot {X}_{r}$ and $dot {M}^{p,d/r}$, when the solution is global for sufficiently small initial data. Furtheremore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions.